ICMI Study 19: Proof and proving in mathematics education

Yet clearly, for mathematicians proof is much more than a sequence of correct steps; it is also and, perhaps most importantly, a sequence of ideas and insights with the goal of mathematical understanding—specifically, understanding why a claim is true. [...]the challenge for educators is to foster the use of mathematical proof as a method to certify not only that something is true but also why it is true. [...]the learning of proof and proving in school mathematics should be developmental and should start in the early grades. How can teachers and mathematics educators use our knowledge about learners’ cognitive development to develop ways of teaching proof that take account of each learner’s growing ways of proving? From a very young age, students show high degrees of ability in reasoning and in justifying their arguments in social situations; however, they do not naturally grasp the concept of mathematical proof and deductive reasoning. [...]educators must help students to reason deductively and to recognize the value of the concept of mathematical proof.